D E V E L O P M E N T A N D A N A LY S I S O F A N E W D G P S AT T I T U D E D E T E R M I N AT I O N M E T H O D F O R M I N I AT U R E S AT E L L I T E S

D E V E L O P M E N T A N D A N A LY S I S O F A N E W D G P S AT T I T U D E D E T E R M I N AT I O N M E T H O D F O R M I N I AT U R E S AT E L L I T E S

s t e f a n o t o r r e s a n

Dipartimento di Ingegneria Industriale

Corso di Laurea Magistrale in Ingegneria Aerospaziale Universitá degli Studi di Padova

A B S T R A C T

In this thesis the concept of attitude determination with phase carrier differential GPS measures for miniature satellites is analysed, with particular attention dedicated to the effects of the short antenna base-line. At first the main features of the GPS and the GPS signal are outlined. The problem of attitude determination is presented, with particular focus on the QUEST algorithm. The concept of phase car-rier DGPS measure is introduced along with the main DGPS-related experiments, such as RADCAL and TOPSAT; the algorithm required to convert the phase difference measure into a vectorized measure is also derived. Two antenna layouts based on a Cubesat standard minia-ture satellite are proposed, with four patch and four helix antennas respectively, and their merits are evaluated through a number of sim-ulations aimed to analyse the satellite availability and the attitude estimation accuracy for each one. Finally the patch antenna layout is identified as the most promising design and its performances com-pared with the current attitude determination and control systems available off-the-shelf for the Cubesat platform.

S O M M A R I O

C O N T E N T S 1 i n t r o d u c t i o n 1 1.1 Attitude determination 1 1.2 Differential GPS 3 1.3 Miniaturized satellites 4 1.4 Motivation 5 1.4.1 Content summary 6 2 g p s ov e r v i e w 9 2.1 GNSS 9

2.2 Overview of the GPS architecture 10 2.3 Trilateration 10

2.4 GPS signal characteristics 12

2.4.1 Pseudo-Random Noise modulations 12 2.4.2 Navigation Message 12

2.4.3 C/A Signal generation 13 2.5 GPS receiver 13

2.5.1 Antenna 13 2.5.2 RF frontend 15 2.5.3 Downconversion 16 2.5.4 Digitalization 16

2.5.5 Baseband signal processing 16 3 c a r r i e r p h a s e d i f f e r e n t i a l g p s 17

3.1 Attitude parametrization 17 3.1.1 Reference frames 17

3.1.2 Parametrization of rotation 18 3.2 Original attitude determination problem 19

3.2.1 Quaternion parametrization 23 3.2.2 Eigenvalues estimation 25 3.3 DGPS concept 29

5.1.1 Propagation 50 5.1.2 Multipath 50

5.1.3 Carrier to noise ratio 51 5.1.4 Antenna phase center 51 5.1.5 Receiver specific error 52 5.2 Simulations 54

5.2.1 Simulation procedure 54 5.2.2 Simulation results 55 6 c o n c l u s i o n s 63

6.1 Design evaluation 63

6.2 Comparison with available miniature satellite ADCS 64 6.3 Future work 65

6.3.1 In-depth receiver simulation 65 6.3.2 Kalman filtering 65

L I S T O F F I G U R E S

Figure 1 The Differential GPS concept 3

Figure 2 nCube2, a one unit Norwegian built Cubesat 5 Figure 3 Structure of the thesis, chapters are color-coded 7 Figure 4 Intersection points of three spheres 11

Figure 5 Modulation of the L1 signal 13 Figure 6 Structure of the L1 signal 14 Figure 7 GPS receiver diagram 14

Figure 8 Typical patch antenna gain pattern 15 Figure 9 Effect of signal reflection on a Right Handed

Circular Polarized antenna, the reflected signal is not received. 15

Figure 10 Choke ring around a GPS antenna, mitigating multipath effects (source: Trimble website). 16 Figure 11 RPY coordinates 17

Figure 12 Body coordinates 18

Figure 13 Speed comparison[1] for robust estimation

(q-method and SVD, top figure) and fast estima-tion algorithms (QUEST and ESOQ, bottom

fig-ure) 22

Figure 14 QUEST algorithm block diagram 28

Figure 15 The phase difference of the signal received from two antennas 29

Figure 16 Concept scheme of two antennas DGPS 30 Figure 17 Integer ambiguity 30

Figure 18 Examples of antenna geometry: on the left they are arranged in a coplanar fashion, non com-patible with equation42 32

Figure 19 GPS satellites position relative to the GPS YUMA almanac, week 761 36

Figure 20 Polar diagram of a generic radiation pattern 38 Figure 21 GPS signal main lobe 38

Figure 22 Patch and helix antennas, the patch antenna is 25x25x4.5 mm 39

Figure 23 The two proposed layouts 39 Figure 24 Patch antenna beamwitdh 40 Figure 25 Helix antenna beamwitdh 41

Figure 26 Satellite visibility history, patch antenna, bw=120◦, alt=300km 43

Figure 28 Cumulative number of visible satellites with different inclinations, patch antenna, bw=160◦, alt=300km 45

Figure 29 Cumulative number of visible satellites, patch antenna, bw=90◦, i=0◦ 45

Figure 30 Cumulative number of visible satellites, patch antenna, bw=120◦, i=0◦ 46

Figure 31 Cumulative number of visible satellites, patch antenna, bw=160◦, i=0◦ 46

Figure 32 Cumulative number of visible satellites, helix antenna, bw=180◦, i=0◦ 47

Figure 33 Cumulative number of visible satellites, helix antenna, bw=260◦, i=0◦ 47

Figure 34 Effect of baseline length on baseline vector es-timation in presence of errors 50

Figure 35 Effect of antenna phase center uncertainty on baseline definition 51

Figure 36 Block diagram of attitude estimation simula-tions 53

Figure 37 Influence of sightline geometry on attitude

ac-curacy 56

Figure 38 Attitude estimation accuracy, patch antenna, bw=120◦, h=300km, noise=2.5mm. The dotted lines are the 3σ bound for attitude error 57 Figure 39 Attitude estimation accuracy, patch antenna, bw=160◦, h=300km, noise=2.5mm. The dotted lines are the 3σ bound for attitude error 58

Figure 40 Average 3σ attitude error, patch antenna, bw=120◦ 59 Figure 41 Average 3σ attitude error, patch antenna, bw=160◦ 60 Figure 42 Average 3σ attitude error, helix antenna, bw=260◦ 61

L I S T O F TA B L E S

Table 1 Attitude determination sources comparison 2 Table 2 RMS attitude error for the RADCAL

experi-ment 4

Table 3 Representation of attitude 20

Table 4 The YUMA entry for satellite 01, week number 761(October 2013) 36

1

1.1 at t i t u d e d e t e r m i nat i o n

Attitude determination is the process of estimating the orientation of an object in space, using informations provided by dedicated instruments. Attitude is expressed by the rotation of a body frame (axis fixed to the object) with respect to a predefined reference frame. Attitude knowl-edge is required throughout all the spacecraft’s mission profiles, in operations such as main engine burn, proper payload orientation (like telecommunication antennas or telescopes) and many housekeeping functions (radiator and solar panel orientation).

To obtain such information many different sensors exist, each one using different reference objects (sun, earth or star trackers and mag-netic field sensors) or exploiting different physics principles (iner-tial measurement units). Sensors that use external reference (all but IMUs) provide the versor of the reference entity in body frame, by knowing the same versor in relative frame it is possible to estimate the rotation of the body frame w.r.t. the reference frame. IMUs usually provide angular rates that are used to estimate the attitude between the instants of acquisition of the other sensors, called "fixes", if re-quired.

An attitude determination system is usually composed by many types of these sensors, to provide redundancy and to complement each other’s pros and cons (summed up in table 1, adapted from Wertz’s Spacecraft Attitude Determination and Control [2, p. 17]). IMUs

are always used because no other sensor has the same peculiar charac-teristics, i.e. continuous measurements and no external sensors; while the rest of the attitude determination system is a mix of other sensors that depend on required performance, resources budget and orbit characteristics.

Table 1: Attitude determination sources comparison

Reference Advantages Disadvantages

Sun Bright, unambiguous, low

power and weight. Usually must be known for solar cells and equipment protection

Subject to eclipses from cen-tral body; 0.5 deg angu-lar diameter viewed from Earth limits accuracy to 1 arc minute

Earth Always available for nearby spacecraft, bright; largely un-ambiguous (may be Moon interference); necessary for many types of sensor and an-tenna coverage

Sensors must be protected from Sun, resolution limited to 0.1 deg because of horizon definition, orbit and attitude strongly coupled

Magnetic field

Economical, low power re-quirements, always available for low altitude spacecraft

Poor resolution (> 0.5 deg), good only near Earth, limited by field strength and model-ing accuracy, orbit and atti-tude strongly coupled, space-craft must be magnetically clean (or inflight calibration is required), sensitive to biases Stars High accuracy (10−3 _deg),

available anywhere in the sky, essentially orbit independent except for velocity aberration

Sensors heavy, complex and expensive. Identification of stars for multiple target sen-sors is complex and time con-suming, usually require sec-ond attitude system for initial attitude estimates

Inertial Requires no external sensors, orbit independent, high accu-racy for limited time intervals, easily done onboard

Figure 1: The Differential GPS concept

1.2 d i f f e r e n t i a l g p s

The Global Positioning System network, fully operative since the early nineties, grants global positioning service not only to ground users but also to spacecraft; the number of visible GPS satellites is reduced as the altitude increases.

While orbit determination is the most obvious application of space-based GPS, attitude determination is also possible[3]. Using multiple

antennas and analysing the carrier phase of the signal received from multiple satellites it is possible to estimate the line of sight vector between the spacecraft and each visible GPS satellite (sightline), then the orientation of the vector connecting two antennas (called baseline) can be reconstructed; using at least three baselines the attitude of the platform can be estimated. This is shown in figure1.

A differential-GPS attitude sensor offers many advantages: • unique platform for orbit position and attitude determination • precise timing available as a free by-product of the measure • small impact on resources (mass, volume and power) • independent from time and orbit position

• external reference measure (unlike IMUs) with a high rate (up to 1 Hz and more)

DGPS attitude sensors have been proposed since the eighties and some spacecraft used this technology, such as RADCAL[3][4] and

GADAC[5] in the early nineties, while TOPSAT[6] and Minnesat[7]

RADCAL (RADar CALibration) is a gravity gradient stabilized small satellite launched in 1993 that was used to calibrate several radar sys-tem of the United State Department of Defence (DoD), it carried four GPS antennas, with baselines around 60cm or less, and two GPS re-ceivers in order to test the concept of DGPS attitude determination.

The data acquired throughout the mission were used to develop and test the first algorithms for integer ambiguity resolution but were not used or processed on-board. The attitude accuracy range was ini-tially around 1◦(1σ) but more accurate estimates were developed, the result is reported in table2(adapted from [4, p. 78])

Table 2: RMS attitude error for the RADCAL experiment

Component Min RMS er-ror [deg] Max RMS er-ror [deg] Average RMS error [deg] Yaw 0.271 0.415 0.317 Pitch 0.457 0.533 0.495 Roll 0.457 0.536 0.496

TOPSAT (Tactical OPerational SATellite) is a micro-satellite demon-strator with a high resolution telescope developed by Surrey Satellite Technology and launched in 2005. As a secondary attitude determi-nation system it carries four GPS patch antennas with a baseline of 0.6m, the spacecraft performed several experiment both in a nadir-pointing attitude and in a pitch/roll off-boresight mode, the attitude estimation was performed on-board with a 1Hz update rate.

The results for this experiments in terms of attitude estimation dis-parity (between the primary ADCS1

and the DGPS) where 0.61◦RMS in roll, 0.71◦RMS in pitch and 1.29◦ RMS in yaw; however the DGPS measure for the yaw angle was considered more accurate than the ADCS one leading to a estimated yaw error of 0.5◦RMS.

1.3 m i n i at u r i z e d s at e l l i t e s

Miniaturized satellites, with the notable example of Cubesats, grant access to the design and construction of operative spacecraft to Uni-versities (such as the nCube2, figure 2 (source: Wikipedia)), small companies and even private users.

Cubesats are composed of one or several units, each one is 10cm x 10cm x 10cm; their small dimensions impose severe constraints to the resource budget allowed to attitude determination, nonetheless

Figure 2: nCube2, a one unit Norwegian built Cubesat

many miniaturized attitude sensors exist.

Miniature satellites have been used to perform Earth remote sens-ing, tethered spacecraft tests, spacecraft subsystem tests and for edu-cational purposes.

While standard GPS receivers have been developed for miniature satellites application, currently no DGPS attitude sensor exist.

1.4 m o t i vat i o n

The purpose of this thesis is to evaluate the performance of a DGPS at-titude sensor for miniature satellite applications, identifying the main sources of error and analysing their influence on the attitude estimate. While the DGPS technology is being researched for more than twenty years, it was limited to isolated experiment.

The advantages presented by this technology suit perfectly a minia-turized application such as Cubesats and the related constraints, how-ever this approach has not been followed before because of the drastic reduction of antenna baseline: while previous DGPS experiments had baselines around 0.6m, a DGPS sensor in a Cubesat is limited to 0.1m; this can reduce considerably the accuracy of the attitude estimation.

In order to evaluate the performances of such miniaturized system it is necessary to analyse both its hardware and software elements and to model the errors that perturb the attitude estimate; these errors are modeled in terms of phase noise, the main DGPS measurement.

not enough for an accurate error estimate, it is possible to formulate the requirements for the next generation receiver and also identify an optimal antenna layout.

1.4.1 Content summary

Chapters 2 and 3 are devoted to an introduction to GPS, attitude de-termination and differential GPS concept; the algorithms for attitude determination with differential GPS phase measures also will be pre-sented in depth.

GPS overview AD overview GPS con-stellation and signal Antenna modelling DGPS AD AD algorithms Availability simu-lations AD sim-ulations DGPS AD system evaluation Overview Insight Simulation Analysis

Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6

2

s u m m a r y

This chapter will present an introduction to the Global Navigation System. At first the different GNSS available or proposed are re-viewed, then a more GPS-focused section will follow with an overview on the GPS architecture.

The concept of trilateration, as well as the bases of GPS signal char-acteristics and a scheme of a GPS receiver, are introduced to give a basic understanding on the GPS function.

The main reference sources used throughout this chapter are Global Positioning System Theory and Applications [8] and Global Positioning

Systems, Inertial Navigation, and Integration[9].

2.1 g n s s

A Global Navigation Satellite System (GNSS) is a combination of satel-lite constellation and ground infrastructure that provides accurate po-sitioning with a global coverage. There are several GNSS systems both operational and under development:

• GPS (Global Positioning System): the first GNSS, developed by the USA for military and civilian use. It uses a semi-synchronous Medium Earth Orbit (MEO) constellation with 32 satellites. • GLONASS (GLObal NAvigation Satellite System): the Russian

equivalent to the GPS, it suffered from incomplete coverage un-til the 2000s. 24 satellites are used in this system.

• Galileo: the European civil-oriented GNSS, currently under de-ployment with four satellites in orbit; to be completed by 2019 (30 satellites constellation).

• BeiDou-2: the upgrade of the Chinese BeiDou-1, composed by satellites in different orbits (3 geosynchronous1

, 5 geostation-ary2

and 27 Medium Earth Orbit) and to be completed by 2020. Other positioning systems have similar concept but offer a limited coverage:

1 The geosynchronous orbit, or GSO, is a circular orbit with a period of exactly one

sidereal day (23h 56m 4s), this is achieved at an altitude of 35, 786km.

2 The geostationary orbit, or GEO, is a particular geosynchronous orbit with no

• IRNASS (Indian Regional Navigation Satellite System): under development, the proposed system uses 7 satellites in both geo-stationary and geosynchronous orbit.

• BeiDou-1: it is owned by China and currently offers a regional coverage.

While different in some aspects, most of these systems share many similarities. Because of its importance and widespread use the GPS will be analysed further in detail, but the same goes for the majority of the GNSS systems worldwide.

2.2 ov e r v i e w o f t h e g p s a r c h i t e c t u r e The Global Positioning System is composed by:

• the Space Segment: 32 satellites arranged in six orbital planes, four satellites for each plane, with an orbital inclination i = 55deg; the planes are evenly spaced with a RAAN (Right As-cension of the Ascending Node) separation of Ω = 60 deg. The orbits radius is approximately 26.600km (altitude of 20.200km) in order to have an orbital period of half a sidereal day (11h 58m 2s), this grants that every satellite repeats the same groundtrack twice a day and at least six satellites are always in view from any point on the surface of the Earth.

• the Control Segment: composed by a number of ground station (1 master, 1 alternate master, 4 ground antennas and 6 monitor stations) spread all over the world; they are operated by the US Air Force and continuously monitor the GPS constellation, track the position of each satellite and update the ephemeris for the navigation message.

• the User Segment: any military, civil or commercial GPS receiver. The GPS reached initial operational capability in 1993 with 24 satel-lites in orbit transmitting the Standard Positioning Service (SPS), in 1995 achieved the full operational capability with the Precise Posi-tioning Service (PPS, available only to military users). By 2000 the civil signal is no longer disturbed, the so-called Selective Availability (SA), granting an accuracy better than 20 meters; future GPS satellites won’t be able to perform SA.

2.3 t r i l at e r at i o n

measure the distance from the satellite to the receiver by knowing the sending and receiving time. The locus of points with the same distance from a reference point is a sphere.

The trilateration is the process of determining the position of a point in space by measuring the distance from reference points of known position. In three-dimensional geometry the intersection of three non-collinear spheres is composed of two points: at least three reference points are therefore needed to compute the position of the unknown point, narrowing it down to two points. This is shown in figure4.

Figure 4: Intersection points of three spheres

Translating this concept to GPS gives the minimum information that the receiver requires in order to determine its own position:

• Satellite position at the instant of transmission • Satellite time at the instant of transmission • Receiver time at the instant of reception

The main source of error in this procedure is the time: at the speed of light an error of 1µs in the time of flight measure becomes a 300m error in range. This is why ultra-stable atomic clocks are used on board the satellites, granting a time accuracy of ±10ns.

2.4 g p s s i g na l c h a r a c t e r i s t i c s

The concept behind any GNSS is code ranging, a technique to obtain a position fix from a number of coded signals.

In the case of the GPS the signal is composed by a high-frequency carrier, a low-frequency Pseudo-Random Noise (PRN) modulation and a even lower-frequency Navigation Message (NM) modulation. While the carrier frequency is single, PRN and NM are unique to each satel-lite, enabling the receiver to acquire and discriminate different signals transmitted on the same carrier. Both PRN and NM are binary mod-ulations. For the GPS the carrier frequencies are (f0= 10.23MHz):

• L1: fL1= 154f0= 1575.42MHz

• L2: fL2= 120f0= 1227.60MHz

2.4.1 Pseudo-Random Noise modulations

The purpose of a PRN code is both to identify the satellite and to provide a sequence to which the receiver can "lock", isolating the message sent from one satellite from the others. There are different Pseudo-Random Noise modulations, both on L1 and L2:

• The Coarse Acquisition (C/A) code modulates L1 and is com-posed by 1023 impulses at a frequency of 0.1f0 = 1.023MHz,

the signal repeats itself every 1ms. This signal is public, mean-ing that the PRN codes are stored inside the GPS receiver in order to make it capable to acquire the signal.

• L2C is similar to C/A, it modulates L2 with 10230 impulses at a frequency of 0.1f0 = 1.023MHz the signal repeats itself every

01ms. This signal is public and transmitted from Block IIR-M satellites and later.

• P(Y) is the encrypted military code on L2, is composed by a very large number of impulses and it repeats itself every 37 weeks • M is a new military code that modulates both L1 and L2, no

information on this code is available. 2.4.2 Navigation Message

Figure 5: Modulation of the L1 signal

2.4.3 C/A Signal generation

It is shown in figures5and6(source: Global Positioning Systems Inertial Navigation and Integration[9]) the structure of the L1 carrier modulated

with the C/A code and the Navigation Message. 2.5 g p s r e c e i v e r

In order to access the information transmitted the encoded GPS sig-nal must be received and acquired properly. The GPS receiver is com-posed by the following elements, that process the signal in cascade:

• Antenna

• RF frontend amplification • Downconversion

• Digitalization

• Baseband signal processing

Figure7shows a block diagram of a generic receiver (source: Global Positioning Systems Inertial Navigation and Integration[9]).

2.5.1 Antenna

Figure 6: Structure of the L1 signal

Figure 7: GPS receiver diagram

much with the angle the low-angle/low-gain signals may interfere with the strongest ones. Therefore the antenna design is a tradeoff be-tween gain pattern and interference. An example of the gain pattern of a patch antenna is shown in figure8(source: Wikipedia).

Figure 8: Typical patch antenna gain pattern

Figure 9: Effect of signal reflection on a Right Handed Circular Polarized antenna, the reflected signal is not received.

creating errors in position and above all attitude estimation. Because every reflection comes with a 180-degrees shift in signal polarization and the GPS signal itself has a right-handed circular polarization (RHCP), a RHCP antenna ignores all the signals coming from one reflection because they are LHCP (9); further reflections have usually lower strength. Alternatively the antenna can be shielded with metal rings of different diameters (choke ring), as shown in figure10. 2.5.2 RF frontend

Figure 10: Choke ring around a GPS antenna, mitigating multipath effects (source: Trimble website).

2.5.3 Downconversion

In order to further amplify the signal its frequency must be lowered to one (single stage) or several (multistage) Intermediate Frequency (IF), this grants benefits from a filtering and Signal-to-Noise Ratio (SNR) aspects.

2.5.4 Digitalization

The Analog-to-Digital Converter (ADC) is used to sample and digital-ize the signal, the frequency of this process must be chosen carefully in order to avoid aliasing and usually is several times the last IF value: without the downconversion of the signal the appropriate sampling frequency will be prohibitive.

2.5.5 Baseband signal processing

3

s u m m a r y

In this chapter various topics related to DGPS attitude determination are presented.

After the review of the various attitude parametrization alterna-tives the attitude determination problem is analysed, focusing on the QUEST algorithm; then the concept of Differential GPS attitude de-termination is presented and the related algorithms derived.

3.1 at t i t u d e pa r a m e t r i z at i o n

For the sake of attitude determination, as introduced in chapter 1, reference versors are required both in reference frame and in body frame. It is useful to define these reference frame, along with the means used to describe a spacecraft’s attitude.

3.1.1 Reference frames

Figure 11 shows the roll-pitch-yaw (RPY) coordinates, used mainly for planet observation (also called nadir-pointing) spacecraft: the Z (yaw) axis is pointed towards nadir, the X (roll) axis is in the direc-tion of the spacecraft’s velocity and the Y (pitch) axis completes the left-handed frame. This reference frame will be used throughout this thesis.

The body frame is any left-handed frame that moves rigidly with the spacecraft, since most spacecrafts are similar to simple three

Figure 12: Body coordinates

mensional bodies the axis direction is used to refer to the spacecraft’s outer panels (figure12).

3.1.2 Parametrization of rotation

The rotations from reference to body frame can be parametrized in many ways (see table3, adapted from Wertz’s Spacecraft Attitude De-termination and Control [2, p. 412]), the most used ones are Euler angles

and quaternions. 3.1.2.1 Euler angles

Euler angles are defined as three successive rotations around the axis of four left-handed orthogonal triads, each triad is the result of the rotation of the previous one around one of its axis. It is common to name each sequence with the number of the axis that are to rotate, i.e. 1-2-3 means that the triad xyz is rotated around x and triad x0y0z0 is obtained, then a rotation about y0leads to x00y00z00, finally a rotation around z00generates XYZ .

The most utilized sequences are 3-1-3 (precession-nutation-spin) for the description of the motion of spinning bodies such as spin-stabilized spacecraft, and 3-2-1 (yaw-pitch-roll) for the representation of attitude of three-axis stabilized spacecraft; for small rotations all the different sequences become similar.

3.1.2.2 Quaternion

A quaternion is a set of four parameters representing a rotation of θ about an axis bX: q =  Q q  =  b Xsin(θ/2) cos(θ/2)  (1) The quaternion is subject to a normalization condition:

qTq =|Q|2+ q2 = 1 (2)

Having only one constraint is one of the advantages of this parametriza-tion, the greatest one being that they have no singularities1

. They are used mainly in attitude determination software for onboard comput-ers.

3.1.2.3 Gibbs vector

Another parametrization than will be used in the subsequent section is the Gibbs vector. It is similar to the quaternion but composed only of three parameters:

Y = bXtan(θ/2) = Q

q (3)

The Gibbs vector becomes infinite when θ = π. It is possible to express the quaternion in terms of the Gibbs vector:

q = _p 1 1 +|Y|2  Y 1  (4) 3.2 o r i g i na l at t i t u d e d e t e r m i nat i o n p r o b l e m

The main task in attitude determination is to find the rotation matrix between two coordinates frame, i.e. vb = Rbava (Rba is the rotation

matrix from frame a to frame b), given a set of n measurements. If b

V_iis the i-th versor in reference frame and cW_ithe i-th versor in body frame, the attitude or rotation matrix A satisfy the relationship:

c

Wi= A bVi for i = 1..n (5)

1 For a deeper discussion of quaternions see Spacecraft Attitude Determination and

Table 3: Representation of attitude

Parame-trization

Advantages Disadvantages Common appli-cations Direction cosine matrix No singularities, no trigonometric func-tions, convenient

product rule for

successive rotations

Six redundant pa-rameters

Analysis

Euler axis and angle

Clear physical inter-pretation One redundant parameter, axis undefined when sin φ = 0, trigono-metric functions Commanding slew manoeuvres Euler angles No redundant pa-rameters, physical interpretation is

clear in some cases

trigonometric func-tions, singularity at some θ, no conve-nient product rule for successive rota-tions

Analytic studies,

input/output,

onboard attitude

control of 3-axis sta-bilized spacecraft Gibbs vec-tor No singularities, no trigonometric func-tions, convenient

product rule for

successive rotations

Infinite for 180 deg rotations

Analytic studies

Quaternion No singularities, no trigonometric

func-tions, convenient

product rule for

successive rotations

One redundant pa-rameter, no obvious physical interpreta-tion

Onboard inertial

However if the measurements are affected by errors the attitude matrix is not unique.

At first only a deterministic solution for pairs of measurements was available, where part of the measurements were discarded and the attitude matrix computed in a non-optimal way; one example be-ing the TRIAD algorithm[10] based on the algebraic solution. Where

more than two measurements are available, such as with star trackers measurements, the algorithm needs to run with every pair of mea-surements and then the attitude matrix is obtained by combining the single results.

In order to find an optimal solution it is useful formulate the prob-lem in terms of a loss function, as proposed by Whaba[11]:

L(A) = 1 2 n X i=1 a_i|Wc_i− A bV_i|2 (6)

The minimization of L(A) is complicated by the six constraints of the attitude matrix A: this is because A is required to be an orthonor-mal matrix, i.e. its rows and column are orthogonal and with unitary norm (AT_{A = AA}T _{= Iwhere I is the identity matrix).}

By parametrizing the attitude matrix with quaternions (q-method) Davenport[12] reduces the constraints from six to one, namely the

normalization of the quaternion; furthermore it success in turning the quadratic loss function into a eigenvalue problem.

The solution of a four-dimensional eigenvalue problem can be com-putational heavy, especially if attitude is rapidly changing and the algorithm has to run several times per second; nonetheless this is an accurate alternative when attitude estimation is seldom needed.

An approximated solution of Whaba’s loss function parametrized with quaternions is proposed by Shuster[10], where the eigenvalues

are found using a Newton-Raphson iterative scheme. This algorithm, named QUEST (QUaternion ESTimator), allows much faster compu-tations while being an optimal solutor.

Other than QUEST other algorithms exist, such as Markley’s SVD[13]

and FOAM[14], Mortari’s ESOQ[15] and ESOQ-2[16]. They differ in

terms of speed and accuracy[1], some of them are suited for

applica-tion when the sensor used have very different accuracies, such as star trackers and sun sensors. Figure 13 present a speed comparison for different algorithms, in terms of number of floating point operations for each run.

following Davenport, then the proper QUEST algorithm will be pre-sented.

3.2.1 Quaternion parametrization

It is possible to write the loss function6as a gain function g(A), then writing the Froebenius norm with the trace operator2

: g(A) = 1 − L(A) = n X i=1 a_iWc_iTA bV_i= tr(ABT) (7)

where B is the attitude profile matrix.

B =

n

X

i=1

aiWc_iVb_iT (8)

As stated earlier the quaternion parametrization is used in order to bring down the number of constraints from six to one; it is possible to write the attitude matrix as a function of the quaternion q:

A(q) = (q2−Q · Q)I + 2QQT + 2qQ (9) Q =     0 −Q3 Q2 Q₃ 0 −Q₁ −Q₂ Q₁ 0     (10)

This allows to write the gain function as:

g(q) = (q2−Q · Q)tr(BT) + 2tr(QQTBT) + 2qtr(QBT) (11) Making use of the definition of quaternion:

where the different quantities are such defined: σ = tr(B) n X i=1 aiWc_i· bV_i (14) S = B + BT = n X i=1 a_i(cW_iVb_iT+ bV_iWc_iT) (15) Z = B − BT = n X i=1 a_i(cW_i× bV_i) (16)

Equation12 must be maximized while being subjected to the nor-malization condition of the quaternion.

The Lagrange multipliers method allows to solve a constrained maximization problem introducing n scalar constants, where n is the number of constraints. The Lagrangian function is then formulated by adding to the original function the constraints:

g(q0) = qTKq − λqqT (17)

λ is the Lagrange multiplier. Maximization is achieved through derivation:

Kq = λq (18)

This is the definition of an eigenvector problem, so q must be an eigenvector of K and λ its related eigenvalue. Substituting equation 18back into equation12leads to

g(q) = qTKq) = λqTq = λ (19)

It is straightforward that the maximum value of g(q) will be ob-tained for the maximum eigenvalue λ. It is therefore proven that the original Whaba’s maximization problem6is reduced to the search of the maximum eigenvalue of the K matrix, or:

3.2.2 Eigenvalues estimation

From the definition of the K matrix 13 and the quaternion 3 it is possible to formulate equation18as a set of two separated equations:

" S − σI Z ZT σ #  Q q  = λ  Q q  (21)    [S − (σ + λ)I]Q + Zq = 0 ZTQ + (σ − λ)q = 0 (22)    Y = Q_q = [(σ + λ)I]−1Z λ = σ +Q_qZ = σ + YZ (23)

Substituting the expression for Y in the very last equation leads to:

λ = σ + ZT 1

[(λ + σ)I − S]Z (24)

As stated before the Gibbs vector becomes infinite for a rotation of π, so this expression is invalid near that rotation: this coincide with the matrix [(λ + σ)I − S] becoming singular.

The goal now is to find an approximate solution of the characteris-tic equation, similar to equation24, expressed with quaternions.

The Cayley-Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation. If M is a square matrix its characteristic equation is:

det|M − λI| = 0 (25)

det     M₁₁− λ M₁₂ M₁₃ M₂₁ M₂₂− λ M₂₃ M31 M32 M33− λ     = = (M11− λ)(M22− λ)(M33− λ) + M12M23M31+ M13M21M32 − (M₂₂− λ)M₃₁M₁₃− (M₁₁− λ)M₂₃M₃₂− (M₃₃− λ)M₂₁M₁₂ = −λ3+ λ2(M₁₁+ M₂₂+ M₃₃) | {z } tr(M) + λ(−M_| 11M22− M22M33− M11M33_{z+ M31M13+ M32M23+ M21M12_}) tr(adj(M)) + M_| 11M22M33+ M12M_{z23M31+ M13M21M32_} det(M) −M₂₂M₃₁M₁₃− M₁₁M₃₂M₂₃− M₃₃M₂₁M₁₂ | {z } det(M) (26)

where the tr is the trace and adj is the classical adjoint (or adju-gate), i.e. the transpose of the cofactor matrix3

.

−λ3+ tr(M)λ2− tr(adj(M))λ +det(M) = 0 (27) Applying the Cayley-Hamilton to the last equation:

−M3+ tr(M)M2− tr(adj(M))M +det(M)I = 0 (28) Defining the following quantities equation28becomes:

σ = tr(M) k = tr(adj(M)) ∆ = det(M) (29)

−λ3+ σλ2− kλ + ∆ = 0 (30)

By the definition of C-H theorem M can be any square matrix, if M = [(λ + σ)I − S]:

[(λ + σ)I − S]−1= γ−1(αI + βS + S2) (31) where

α = λ2− σ2+ k β = λ − σ γ = (λ + σ)α − ∆ (32)

3 The elements of the cofactor matrix are C_ij = (−1)i+jm_ij, where m_ij is the minor

Substituting equation31 back into the expression for the eigenval-ues derived earlier (equation24) leads to:

λ4− (a + b)λ2− cλ + (ab + cσ − d) = 0 (33)

a = σ2− K b = σ2+ ZTZ c = ∆ + ZTSZ d = ZTS2Z (34) Finally a simplified equation for the eigenvalues is derived. Equa-tion 33 can be solved with an iterative method, such as Newton-Raphson method. This is particularly convenient because the method needs a starting value for λ, but λ = g(A) (equation 19) and since g(A)is to be maximized, the initial guess can be λmax= 1.

Once the value for λmax is obtained the only thing left is

calculat-ing the attitude quaternion. Again from equation31and4, substitut-ing λ = λmax: q_opt = _p 1 γ2₊|X|2  X γ  (35) where X = (αI + βS + S2)Z (36)

Data as-signment Weight vector Attitude profile matrix Matrixes and operators Character-istic equation parameters Newton-Raphson Optimal quaternion b V, cW, σ a_i= σ2T OT σ2 i = Pn i=1(σ2i) σ2 i B =Pn_i=1a_iWc_iVb_iT S = B + BT Z = B − BT σ = tr(B) ∆ = det(S) Cij= (−1)ijdet(Am,ij) k = tr(CT) a = σ2− k b = σ2+ ZTZ c = ∆ + ZTSZ d = ZTS2Z f(λ) = λ4− (a + b)λ2− cλ + (ab + cσ − d) f0(λ) = 4λ3− 2(a + b)λ − c α = λ2− σ2+ k β = λ − σ γ = (λ + σ)α − ∆ X = (αI + βS + S2)Z q_opt = _p 1 γ2_+|X|2  X γ 

Figure 15: The phase difference of the signal received from two antennas

3.3 d g p s c o n c e p t

Differential GPS techniques make use of multiple receivers in order to augment the precision of the reconstructed position and are the focal point in GPS attitude determination.

Using an appropriate tracking loop it is possible to measure the phase of the GPS signal carrier, the difference between measurement taken by different antennas is proportional to their relative range with respect to the observed GPS satellite.

The phase difference between the signal received from two anten-nas is outlined in figure 15; from the figure it can be noted easily that the phase difference ∆φ is comparable to a length, the distance traveled is r = λ∆φ where λ is the wavelength of the carrier.

If the relative position of the antennas is fixed, i.e. they are mounted on a rigid body, the relative range is also proportional with the angle between the vector connecting the antennas, or baseline, b and the GPS satellite line of sight s.

The phase difference measure ∆φ0 _{is related to the relative range}

∆rand the angle θ by:

∆r = blcos θ = λ

∆φ0

2π (37)

Figure 16: Concept scheme of two antennas DGPS

Figure 17: Integer ambiguity

receiving antennas.

If the baseline length blis longer than the wavelength of the carrier

It is possible, making use versor notation for the baseline and sight-line versors b and s with no integer ambiguity, to define the normal-ized phase difference measurement ∆φ :

∆φ = λ∆φ

0

2πb_l =b

T

s (40)

From this definition it is evident that the normalized phase differ-ence ∆φ is the direction cosine of the two versors; this relationship is valid for all the reference frames, provided that the same frame is used for expressing both b and s.

3.4 g p s at t i t u d e d e t e r m i nat i o n

The GPS measurements introduce another aspect in attitude deter-mination: the phase difference is not a vector but a scalar, and vec-torized measurements are needed in order to use attitude estimation algorithm such as QUEST.

In the conversion of phase difference measure to vectorized mea-sures equation40is a central point, following Crassidis and Markley[17].

The two versors groups, sightlines and baselines, can be expressed in both the reference and the body frame; of these four quantities two are known:

• sightlines in reference frame sb, determined from knowledge of

the spacecraft and GPS constellation orbit positions • baselines in body frame br, known from spacecraft design

The other two are to be obtained by converting the scalar phase differences; the problem can be stated in two ways, both of them taking the steps from equation ∆φ = bTs:

1. determining sightlines in body frame s_bleads to ∆φ = bT

bsb

2. determining baselines in reference frame b_rleads to ∆φ = bT

rsr

Figure 18: Examples of antenna geometry: on the left they are arranged in a coplanar fashion, non compatible with equation42

3.4.1 Sightlines in body frame

The first cost function, expressed in body coordinates, is:

J_j(sj) = 1 2 m X i=1 1 σ2_ij(∆φij−b T isj)2 for j = 1..n (41)

where n is the number of observed GPS satellites (number of sight-lines), m is the number of available baselines and σ2

ijis the covariance

of the measurement process. The inverse of the covariance _σ12 ij is the

equivalent of the weight ai in equation6. Minimization of this cost function leads to:

sj= M−1_j yj (42) where M_j = m X i=1 1 σ2bib T i yj = m X i=1 1 σ2∆φijbi for j = 1..n (43)

The major consequence of this approach is that at least three non-coplanar baselines are required: this has a great influence on the system’s design, as shown in figure18.

3.4.2 Baselines in reference frame

Developing the same solution expressing the baselines in reference frame leads to the following equations:

It is worth pointing out that now the sum is over the n sightlines, while all these equation are in reference frame. Minimization leads to: bi= N−1_i zi (45) where n_i= n X j=1 1 σ2sjs T j zi= X j=1 n 1 σ2∆φijsj for i = 1..m (46)

As anticipated before, the constraint posed by equation45 is that at least three non-coplanar sightlines are required. This is a less restrain-ing condition than the first one, since the orbital position of the GPS constellation, with satellite spread over six orbital planes, renders this event impossible.

4

s u m m a r y

The first concern regarding DGPS in space application is signal avail-ability: while the GPS constellation is designed to provide a minimum number of six satellites visible from any point on Earth surface, this number drops as the altitude of the receiving platform rises.

Two different antenna layouts will be evaluated for different orbit altitudes and antenna gain characteristics.

4.1 s i m u l at i o n pa r a m e t e r s

The number of visible GPS satellites is a major factor in DGPS attitude determination; a minimum number of three is the requirement for the second attitude determination algorithm shown in the previous chap-ter. More than three satellites will improve the attitude estimation, so the number of available satellites is a first measure of the system’s overall accuracy.

The signal availability is influenced by the following factors: • orbit geometry

• GPS antenna emission power and radiation pattern • receiving antenna radiation pattern and layout geometry 4.1.1 Orbit geometry

GPS satellites are positioned along six orbital planes, each one with an inclination of 55◦ and with a separation of 60◦. Originally four satellites per plane were operative for a total of 24, but this number has increased to 32 to improve precision and availability.

The orbit altitude is 20200km to grant a semi-synchronous period, i.e. the orbit period is half a sidereal day.

The GPS almanac data are available on the Navigation Center site of U.S. Department for Homeland Security1

, for every GPS week and in various formats. An example of an entry of the YUMA almanac is shown in table 4. The satellites position for the same almanac data are represented in figure19.

Table 4: The YUMA entry for satellite 01, week number 761 (October 2013)

******** Week 761 almanac for PRN-01 ********

ID: 01

Health: 000

Eccentricity: 0.2826213837E-002

Time of Applicability(s): 319488.0000

Orbital Inclination(rad): 0.9609934236

Rate of Right Ascen(r/s): -0.7760323249E-008

SQRT(A) (m 1/2): 5153.638184

Right Ascen at Week(rad): -0.5683245271E+000

Argument of Perigee(rad): 0.332184364

Mean Anom(rad): 0.1340404472E+001

Af0(s): 0.5722045898E-005

Af1(s/s): 0.0000000000E+000

week: 761

4.1.2 GPS antenna

The next subjects of analysis are GPS spacecraft’s antenna power and radiation pattern. While the former is not an issue, due to the fact that an orbiting spacecraft is usually closer to the signal source than a ground user and it doesn’t have to deal with atmospheric distur-bances, the latter becomes more and more influential as the altitude increase. Extensive analysis of the link budget for a spacecraft-based GPS receiver are performed in [? ] and show that the available power on orbit is equivalent or better than on the ground.

The gain of an antenna, i.e. the power of the signal of the antenna compared with the power irradiated from an isotropic antenna, is a function of the direction. A generic radiation pattern is shown in figure 20 and its main element outlined: the signal amplification is measured in decibel (dB), the main lobe corresponds to the main emitting/receiving direction of the antenna and the half-power band-width is the arc inside which the gain is inside a −3dB range with respect to the maximum gain, this corresponds to a half-power con-dition [18].

While the specific characteristics of the antennas depend on the type of GPS satellite, some are very similar. The antenna emission pattern is designed in order to grant an uniform power throughout all the visible Earth surface, the irradiation towards space is minimized and often due to side lobes.

The semi aperture of the main lobe for the L1 signal is∼ 21.3◦[19]

and this value is used as a cut-off for the simulation, meaning that any signal from the side lobes is going to be ignored.

It is evident from figure 21 that this is not a factor for low earth orbits satellites, the range of altitude from 0 to 3000 km being covered; in order to evaluate GPS availability for higher orbits such analysis is required[20].

4.1.3 Receiving antenna

Two types of antennas are considered: patch and helix (see figure 22), each antenna type corresponds to a different antenna layout that is designed to take advantage of each antenna’s characteristics. The layouts are:

• four patch antennas mounted on the same panel (figure23a) • four helix antennas mounted on the edges of the spacecraft

Figure 20: Polar diagram of a generic radiation pattern

Figure 22: Patch and helix antennas, the patch antenna is 25x25x4.5 mm

(a) Patch (b) Helix

Figure 23: The two proposed layouts

In order to maximize the visibility the antennas are pointed to the zenith panel (−Z with respect to figure 12), the spacecraft is consid-ered nadir-pointing.

4.1.3.1 Patch

Patch antennas are usually more economic and have higher gain, however the gain decreases for signals with low incidence angle and the gain pattern itself is very sensitive to the extension and shape of the mounting surface.

This layout is the simplest one from a mechanical point of view and is the most compatible with the Cubesat design. Mounting the antenna on a large surface tends to raise the gain near the zenith and create a more directional radiation pattern, while the opposite is true for smaller surfaces.

Figure 24: Patch antenna beamwitdh

because its shape is a result of a combination of the two opposite ef-fects described previously [21].

In order to achieve a meaningful result the radiation pattern is modelled as a step function, which is 1 (signal received) inside the detection threshold and 0 (signal ignored) outside. While this is a rather simplistic approach, a more in-depth simulation of the link would have required not only precise information about the layout’s radiative characteristics, that due to the effects explained before are obtainable only with tests of the chosen antennas and surface combi-nation, but also a complete implementation of a GPS receiver in all its functions [22]. This approach is also used for the attitude

perfor-mance evaluation of the RADCAL experiment[4], and can give some

meaningful insight on the layout design choices and also a require-ment prediction for the complete DGPS sensor.

For the patch antenna the detection threshold (beamwidth) is set to different values throughout the simulations: 90◦ (worst case), 120◦ (mid case) and 160◦(best case). The 180◦case has not been considered due to antenna to antenna geometric masking.

4.1.3.2 Helix

Helix antennas have lower gain but their gain pattern is more uni-form; an helix antenna is often polarized: as introduced before this helps in the rejection of multipath signals.

Helix antennas are less influenced by the ground plane [21], the

proposed layout is intended to take advantage of their rather omni-directional gain pattern to acquire a greater number of GPS signals; however it doesn’t fit the Cubesat standards as well as the previous design and the helix antenna itself has a greater geometric envelop than the more compact patch antenna.

Figure 25: Helix antenna beamwitdh

of view; the latter case is useful to evaluate the performance of this layout as if patch antenna were used.

4.2 s i m u l at i o n r e s u lt s

The simulations involve a 24 hours period with a time discretization of 10 s, with the GPS ephemeris data taken from the YUMA almanac of week 761 (see table 19). The spacecraft’s orbits considered are cir-cular, with an altitude ranging from 300 to 3000 km; the orbital incli-nation is 0◦but this is irrelevant for this simulation, as will be shown later. The spacecraft is nadir-pointing and the GPS antenna are posi-tioned in the −Z (zenith) direction.

The results are presented in terms of:

• number of visible satellites (time history and cumulative graph over 24 hours)

• availability windows and average availability time

At first some general remarks about the DGPS system are pre-sented, with particular respect to the receiver performance require-ments that the space application poses; then the single layouts analy-sis are detailed.

4.2.1 General considerations

Figure 26 shows a representation of the intervals of time when each GPS satellite is visible by the antenna set. The number on the right of each interval is its duration in seconds.

While figure26 displays the results for a single simulation, tables 5 and 6 summarise the average satellite availability time for all the simulations performed.

This is an important indication because the receiver’s tracking soft-ware needs to be designed in order to acquire the GPS signal in a short period of time, the average satellite availability time ranging from 10 to 30 minutes for a satellite orbiting at an altitude of 300 km. It has been proved [23] that using specifically designed algorithm

circuit it is possible to lower the acquisition time from several tens of minutes to three minutes or less; this is a key aspect in order to exploit at best the available signals.

Figure27represents the time history of the number of visible satel-lites. The graph is very "jumpy", this means that a lot of different satellites come quickly in and out of view: the receiving hardware and software need to be fast and flexible in order to manage the ac-quisition process; again from [23] the use of FPGA technology allows

a greater deal of flexibility than standard integrated circuits.

Inclination is not an important factor, this is evident from figure28 and is an expected result because of the inherent design of the GPS constellation. All the simulations are therefore performed with i=0◦. 4.2.2 Patch

Figures29,30and31summarize the number of visible satellites in a cumulative graph, one for each antenna beamwidth.

The generalized tendency is that higher orbits have a lower number of visible satellites, so the use of DGPS becomes more and more diffi-cult increasing the altitude; moreover the 90◦beamwidth case (figure 29) is not promising because of the possibility that least than three satellites are into view: in these periods of time, a total of 6 hours per day at 300 km and 12 hours at 3000 km, the DPGS attitude measure is not available.

Thus a patch antenna beamwidth of 120◦or more is recommended in order to achieve an attitude determination solution for the major part of an orbit.

4.2.3 Helix

The helix antenna layout will be now analyzed. From table6and fig-ure 32 is evident that the 180◦ beamwidth is not a viable solution, because of the low number of the visible satellites. The other configu-ration (figure33) has very similar results with the 160◦patch antenna: while the helix antenna has a far greater beamwidth than the patch antenna, the need for the satellite to be visible by all the antennas renders irrelevant this advantage.

Table 5: Average availability time, patch antenna

Alt ↓ Beam → 90◦ 120◦ 160◦

300 km 13.1 min 18.4 min 30.3 min 600 km 13.5 min 19.4 min 31.9 min 1000 km 14.6 min 21.1 min 34.1 min 2000 km 16.1 min 23.4 min 39.3 min 3000 km 17.1 min 26.7 min 44.1 min

Table 6: Average availability time, helix antenna

Alt ↓ Beam ang → 180◦ 260◦

300 km 9.9 min 29.7 min 600 km 10.3 min 31.3 min 1000 km 11.0 min 33.5 min 2000 km 12.0 min 38.8 min 3000 km 12.3 min 43.7 min 0 5 10 15 20 25 2 3 4 5 6 7 8 time [hrs] n° visible satellites

0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 8 n° visible satellites total time [hrs] i=0° i=45° i=90°

Figure 28: Cumulative number of visible satellites with different inclina-tions, patch antenna, bw=160◦, alt=300km

0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 n° visible satellites total time [hrs] 300 km 600 km 1000 km 2000 km 3000 km

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 n° visible satellites total time [hrs] 300 km 600 km 1000 km 2000 km 3000 km

Figure 30: Cumulative number of visible satellites, patch antenna, bw=120◦, i=0◦ 6 8 10 12 14 16 1 2 3 4 5 6 7 n° visible satellites total time [hrs] 300 km 600 km 1000 km 2000 km 3000 km

0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 n° visible satellites total time [hrs] 300 km 600 km 1000 km 2000 km 3000 km

Figure 32: Cumulative number of visible satellites, helix antenna, bw=180◦, i=0◦ 6 8 10 12 14 16 0 1 2 3 4 5 6 n° visible satellites total time [hrs] 300 km 600 km 1000 km 2000 km 3000 km

5

s u m m a r y

After having addressed the signal availability problem in the previous chapter the attitude estimation accuracy of the proposed system will be analysed.

The main error sources will be described and the ones most rele-vant to the problem modeled in the simulation software. After that the description of the simulation and an outline of its results will follow.

5.1 e r r o r s o u r c e s

The first step in accuracy estimation is to identify, quantify and model the main error sources in the signal processing. The analysis follows Cohen[3] and takes advantage of some peculiarities of the differential

GPS technique in order to neglect some of them. The error sources are:

• propagation • multipath

• carrier to noise ratio • antenna phase center • receiver

Before detailing all these it is important to stress that the baseline length has a great influence in the errors propagation. If the cumu-lative error from all the sources is represented as a circle around the geometric location of the antenna this effect is clear: a shorter baseline implies a greater error, as shown in figure34.

This effect is not avoidable and the only way to mitigate it is to try and minimize the cumulative error, that equals to reduce the diame-ter of the error circle.

Figure 34: Effect of baseline length on baseline vector estimation in presence of errors

5.1.1 Propagation

Due to the fact that the GPS signals propagate through the atmo-sphere and the ionoatmo-sphere they are subject to the Snell’s Law and then follow a curved path rather than the expected straight-line one.

Since in the analysed application the antenna is at altitude of at least 300km these effects are not a great factor and they are not mod-elled.

Furthermore the very short baseline makes the signal arriving to the different antennas travel virtually the same path, so the distur-bances are equal and they cancel themselves in differential measures. 5.1.2 Multipath

Multipath error comes from the multiple reflections of the incom-ing GPS signal on surfaces around the antenna, these reflected sig-nals have different paths than the original signal and thus the carrier phase measured from them is different from the one on the original signal. This introduces error and a method is to be defined in order to discriminate the original signal from the reflections.

The fist way to reduce multipath is to place the antenna in such a way that it has clear field of view of the sky, however this is not always possible and in a complex structure such a spacecraft there can be a number of protruding elements, such communication anten-nas, gravity stabilization booms and solar panels, that can provide a reflecting surface for the signal.

One advantage of the Cubesat is that it has a very simple and com-pact design, usually the solar panels are not deployable and therefore there aren’t structures that can cause multipath. The only remaining multipath effect can arise with low-incidence signals because the an-tennas are protruding from the panel, this can be addressed with the following considerations.

Figure 35: Effect of antenna phase center uncertainty on baseline definition

by making use of a polarized antenna, i.e. an antenna that is able to receive only right- or left-handed polarized signal, it is possible to ignore the noise coming from first reflection.

Moreover each reflection implies a decrease of the signal’s power, so the original signal is always the one with the highest power: this is useful for discriminating the correct signal[24].

Multipath noise is not easily quantifiable due to its nature: it heav-ily depends on layout geometry and radiation pattern of the antenna group. Cohen [3] states that a differential range error of 5mm RMS

models conservatively multipath for a typical application, this can be reduced by an order of magnitude by calibration of the antenna layout.

5.1.3 Carrier to noise ratio

This is the main source of error and it’s closely related with the preci-sion of the carrier phase tracking loop. Basically the higher this value (the higher the strength of the information with respect to the back-ground noise) the easier will be for the receiver to track accurately the carrier phase. This accuracy is modeled as a random noise (uniform distribution) in the carrier phase measure, expressed either in mm or in deg; Lu [25] identified a value of 1mm RMS as a requirement for

a DGPS attitude determination receiver. 5.1.4 Antenna phase center

This error is caused by the geometric extension of the antenna. In both in the simulation and the real attitude determination estimation the phase measurement refers to the geometric phase center of the antenna; the phase center in the actual antenna layout is not going to be in the center of the antenna neither its position is going to be stable, as it is usually function of the incidence angle of the signal. Figure35illustrates the problem for two patch antennas.

The uncertainty related to this phenomenon is modeled again as a random noise in the carrier phase measurement. In [26] the error is

error for a single antenna by generating a lookup table of the phase center shift as a function of the incidence angle[7].

5.1.5 Receiver specific error

Within this denomination are grouped a number of phenomena that take place inside the receiving equipment or are related with it. This includes:

• Cross-talk: the electromagnetic interferences can be influential because of the several high gain stages, the effects of noise in the early stages can be increased exponentially; this can be con-trolled with proper design of the receiver and is not modelled. • Line bias: since the electrical informations move at a constant

speed, i.e. the speed of light in their medium, the phase of each antenna is delayed by a time proportional to the length of the cable from the antenna to the receiver; this is a systematic error and can be calibrated before use (this remains a factor if the ca-ble is exposed to wide thermal cycles that can alter significantly its length).

• Local oscillator bias: this error is due to the use of different clocks inside the receiver (e.g. one for each channel, each chan-nel can track the signal from one satellite), since the phase mea-sure must be taken at exactly the same time for every channel the time difference from non synced clocks translates directly in false phase difference. By making use of a single clock and multiplexing the channels[3] this error is canceled.

• Inter-channel bias: this is similar to line bias but it is caused by different paths for each channel, using a multiplexed architec-ture this is again canceled.

While these errors are not modeled in the simulation they gave some insight in the inner working of the GPS receiver and on the many calibrations that needs to be carried out on the system, as well as the techniques that are used to reduce the error’s impact.

Table 7: Modeled error sources and average values

Error source Average value Multipath 0.5 ÷ 5mm

C/No 1mm

GPS ephemeris Cubesat orbit Orbital pr opagator GPS state v ectors

Cubesat state vector Antenna _layout

and

pr

operties _{Cubesat attitude}

Signal av ailability and phase measur es Antenna pointing Err or models _Attitude

estimation algorithms Estimated attitude Attitude

5.2 s i m u l at i o n s

The simulation structure is similar to the one used in the previous analysis but with the addition of the attitude estimation procedure.

Table7 sums up the results of the previous paragraphs and is the base for the error input in simulations. The single effects are com-bined in a single value, this perturbs the exact phase difference mea-sure.

5.2.1 Simulation procedure

The basic configurations analysed are the same than the previous chapter, but some are discarded according to the visibility simula-tions results. For each case three error magnitude are analysed, rep-resenting a typical, best or worst case scenario.

• The typical case (2.5mm RMS) is obtained from table7and is rep-resentative of the performances of current available hardware with thorough calibration.

• The best case (1mm RMS) has been chosen arbitrarily in order to provide a perspective of the potentiality of next generation hardware.

• The worst case (7mm RMS) is again taken from table7but with poor or no calibration.

These RMS values are used to generate an uniform distribution interval with amplitude a = RMS2√3, this is used as the phase noise interval in the simulations.

The antenna layout considered are 120◦ and 160◦ for patch and 260◦for helix, having discarded the 90◦ patch and the 180◦ helix; the altitude is ranging from 300 to 3000 km.

Figure 36 shows the structure of the simulation procedure in a block diagram. In particular the attitude determination algorithm fol-lows closely the one presented in chapter3and has two steps:

• conversion from phase differences to vectorized measures • application of the QUEST algorithm to estimate the attitude

quaternion from the vectorized measures

The output of the QUEST algorithm is then converted from quater-nions to Yaw-Pitch-Roll angles.

errors, with a uniform distribution random noise and 1000 runs for every iteration. This allows to calculate the 3σ attitude error (3 times the RMS attitude error).

Two kinds of simulation has been carried out with different pur-poses:

• a short-timespan analysis that aims to evaluate the performances of the algorithm as a real-time attitude estimator. The time step is 1 second, equivalent of the frequency of a typical embedded GPS receiver (the time required to process the signal and per-form attitude estimation), for a total time of 2 hours.

• a longer analysis in order to characterize the overall accuracy of the system, for a total time of 24 hours and a time step of 10 seconds.

5.2.2 Simulation results 5.2.2.1 Patch

The first plot (figure 38) represent the attitude estimation results for the three attitude angles for one of the analysed cases. The yaw, pitch and roll angles error are obtained by subtracting the estimated values from the nominal values; also the number of available GPS satellites and the 3σ bounds for the attitude estimation are given.

On this figure several considerations can arise:

• It is clear that when the number of available measures (n) is less than 3 the attitude algorithm can’t provide an attitude estima-tion, such as in the period between 0.6 and 0.8 hour

• The attitude estimation accuracy is strongly dependent on n: when n = 3 the 3σ bound is as high as 5◦, while for n = 8 the same is less than 1◦

• The direction of the sightline versors is also influential: around 1.4 and 1.6 hours the 3σ bound for the roll and yaw angles is much higher than the one for the pitch error.

From a similar figure another aspect that characterize the DGPS attitude estimation is evident: the yaw accuracy is generally better than the pitch and roll accuracy.

(a) Best for roll and pitch (b) Best for yaw

Figure 37: Influence of sightline geometry on attitude accuracy

This result is consistent with the attitude accuracy estimation of the RADCAL satellite, performed in [4] and outlined briefly in table2.

Comparing figures38 and39 it can be noted that, while the accu-racy difference varies greatly when passing from 3 to 4 visible satel-lites, the visibility of more than 10 doesn’t offer the same advantage.

Figures 40 and 41 present the results for the patch antenna lay-out with two different bandwidth, respectively 120◦ and 160◦. The bw=120◦ at an altitude h=3000km has not been considered because of the very low number of GPS satellites available.

5.2.2.2 Helix

The only case considered for the helix antenna layout is the bw=260◦, the other one (180◦) having too few available satellites as shown in the previous analysis in chapter 4.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −5 0 5

yaw error [deg]

yaw 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −5 0 5

pitch error [deg]

pitch 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −5 0 5

roll error [deg]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−2 yaw error [deg]_{0 2}

yaw 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−2pitch error [deg]_{0 2}

pitch 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−2 roll error [deg]_{0 2}

yaw pitch roll 0 1 2 3 4 5 6 RMS error [deg] altitude = 300 km 1mm 2.5mm 7mm

yaw pitch roll 0 1 2 3 4 5 6 RMS error [deg] altitude = 600 km

yaw pitch roll 0 1 2 3 4 5 6 RMS error [deg] altitude = 1000 km

yaw pitch roll 0 1 2 3 4 5 6 RMS error [deg] altitude = 2000 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 300 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 600 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 1000 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 2000 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 3000 km 1mm 2.5mm 7mm

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 300 km 1mm 2.5mm 7mm yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 600 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 1000 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 2000 km

yaw pitch roll 0 1 2 3 4 RMS error [deg] altitude = 3000 km